Annals of Applied Probability

Randomly growing braid on three strands and the manta ray

Jean Mairesse and Frédéric Mathéus

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Consider the braid group B3=〈a, b|aba=bab〉 and the nearest neighbor random walk defined by a probability ν with support {a, a−1, b, b−1}. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

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Ann. Appl. Probab., Volume 17, Number 2 (2007), 502-536.

First available in Project Euclid: 19 March 2007

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Primary: 20F36: Braid groups; Artin groups 20F69: Asymptotic properties of groups 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)

Braid group B_3 random walk drift harmonic measure dihedral Artin group


Mairesse, Jean; Mathéus, Frédéric. Randomly growing braid on three strands and the manta ray. Ann. Appl. Probab. 17 (2007), no. 2, 502--536. doi:10.1214/105051606000000754.

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  • Aliprantis, C. and Border, K. (1999). Infinite-Dimensional Analysis: A Hitchhiker's Guide, 2nd ed. Springer, Berlin.
  • Bangert, P., Berger, M. and Prandi, R. (2002). In search of minimal random braid configurations. J. Phys. A 35 43–59.
  • Berger, M. (1994). Minimum crossing numbers for $3$-braids. J. Phys. A 27 6205–6213.
  • Berstel, J. and Reutenauer, C. (1988). Rational Series and Their Languages. Springer, Berlin.
  • Dehornoy, P. (2002). Groupes de Garside. Ann. Sci. École Norm. Sup. (4) 35 267–306.
  • Derriennic, Y. (1980). Quelques applications du théorème ergodique sous-additif. Astérisque 74 183–201.
  • Dynkin, E. and Malyutov, M. (1961). Random walk on groups with a finite number of generators. Sov. Math. Dokl. 2 399–402.
  • Eilenberg, S. (1974). Automata, Languages and Machines. A. Academic Press, New York.
  • Epstein, D., Cannon, J., Holt, D., Levy, S., Paterson, M. and Thurston, W. (1992). Word Processing in Groups. Jones and Bartlett, Boston.
  • Garside, F. (1969). The braid groups and other groups. Quart. J. Math. Oxford Ser. (2) 20 235–254.
  • Gromov, M. (1987). Hyperbolic groups. In Essays in Group Theory. Math. Sci. Res. Inst. Publ. 8 75–263. Springer, Berlin.
  • Guivarc'h, Y. (1980). Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. Astérisque 74 47–98.
  • Kaimanovich, V. (1996). Boundaries of invariant Markov operators: The identification problem. In Ergodic Theory of $\mathbfZ^d$ Actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser. 228 127–176. Cambridge Univ. Press.
  • Kaimanovich, V. (2000). The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152 659–692.
  • Ledrappier, F. (2001). Some asymptotic properties of random walks on free groups. In Topics in Probability and Lie Groups: Boundary Theory (J. Taylor, ed.) 117–152. CRM Proc. Lect. Notes 28. Amer. Math. Soc., Providence, RI.
  • Loynes, R. (1962). The stability of a queue with non-independent interarrival and service times. Proc. Cambridge Philos. Soc. 58 497–520.
  • Lyons, R., Pemantle, R. and Peres, Y. (1997). Unsolved problems concerning random walks on trees. In Classical and Modern Branching Processes 84. IMA Vol. Math. Appl. 223–237. Springer, New York.
  • Lyons, R. and Peres, Y. (2006). Probability on trees and networks. To appear.
  • Mairesse, J. (2005). Random walks on groups and monoids with a Markovian harmonic measure. Electron. J. Probab. 10 1417–1441.
  • Mairesse, J. and Mathéus, F. (2004). Random walks on groups with a tree-like Cayley graph. In Mathematics and Computer Science. III. Algorithms, Trees, Combinatorics and Probabilities. Trends in Mathematics 445–460. Birkhäuser, Basel.
  • Mairesse, J. and Mathéus, F. (2006). Random walks on free products of cyclic groups. J. Lond. Math. Soc. Available at arXiv:math.PR/0509211 and arXiv:math.PR/0509208 (appendix).
  • Mairesse, J. and Mathéus, F. (2006). Growth series for Artin groups of dihedral type. Internat. J. Algebra Comput. 16 1087–1107.
  • Mairesse, J. and Mathéus, F. (2005). Appendix to the paper “Randomly growing braid on three strands and the manta ray.” Available at arXiv:math.PR/0512391.
  • Propp, J. and Wilson, D. (1996). Exact sampling with coupled markov chains. Random Structures Algorithms 9 223–252.
  • Nagnibeda, T. and Woess, W. (2002). Random walks on trees with finitely many cone types. J. Theoret. Probab. 15 383–422.
  • Nechaev, S. (1996). Statistics of Knots and Entangled Random Walks. World Scientific, River Edge, NJ.
  • Nechaev, S. and Voituriez, R. (2003). Random walks on three-strand braids and on related hyperbolic groups. J. Phys. A 36 43–66.
  • Picantin, M. (2000). Petits groupes Gaussiens. Ph.D. thesis, Univ. Caen.
  • Sawyer, S. and Steger, T. (1987). The rate of escape for anisotropic random walks in a tree. Probab. Theory Related Fields 76 207–230.
  • Vershik, A., Nechaev, S. and Bikbov, R. (2000). Statistical properties of locally free groups with applications to braid groups and growth of random heaps. Comm. Math. Phys. 212 469–501.
  • Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press.